Question: Simplify; express your answer in exponential form. Assume $q\neq 0, a\neq 0$. $\dfrac{{(q)^{-3}}}{{(q^{-4}a^{2})^{2}}}$
To start, try working on the numerator and the denominator independently. In the numerator, we have ${q}$ to the exponent ${-3}$ . Now ${1 \times -3 = -3}$ , so ${(q)^{-3} = q^{-3}}$ In the denominator, we can use the distributive property of exponents. ${(q^{-4}a^{2})^{2} = (q^{-4})^{2}(a^{2})^{2}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(q)^{-3}}}{{(q^{-4}a^{2})^{2}}} = \dfrac{{q^{-3}}}{{q^{-8}a^{4}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{-3}}}{{q^{-8}a^{4}}} = \dfrac{{q^{-3}}}{{q^{-8}}} \cdot \dfrac{{1}}{{a^{4}}} = q^{{-3} - {(-8)}} \cdot a^{- {4}} = q^{5}a^{-4}$.